Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case
01 Aug 2007-
TL;DR: In this paper, a greedy algorithm called Orthogonal Matching Pursuit (OMP) was proposed to recover a signal with m nonzero entries in dimension 1 given O(m n d) random linear measurements of that signal.
Abstract: This report demonstrates theoretically and empirically that a greedy algorithm called
Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension
d given O(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.
Citations
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08 Aug 2019
TL;DR: A comprehensive overview and analysis of the most recent research in machine learning principles, algorithms, descriptors, and databases in materials science, and proposes solutions and future research paths for various challenges in computational materials science.
Abstract: One of the most exciting tools that have entered the material science toolbox in recent years is machine learning. This collection of statistical methods has already proved to be capable of considerably speeding up both fundamental and applied research. At present, we are witnessing an explosion of works that develop and apply machine learning to solid-state systems. We provide a comprehensive overview and analysis of the most recent research in this topic. As a starting point, we introduce machine learning principles, algorithms, descriptors, and databases in materials science. We continue with the description of different machine learning approaches for the discovery of stable materials and the prediction of their crystal structure. Then we discuss research in numerous quantitative structure–property relationships and various approaches for the replacement of first-principle methods by machine learning. We review how active learning and surrogate-based optimization can be applied to improve the rational design process and related examples of applications. Two major questions are always the interpretability of and the physical understanding gained from machine learning models. We consider therefore the different facets of interpretability and their importance in materials science. Finally, we propose solutions and future research paths for various challenges in computational materials science.
1,301 citations
01 Dec 2013
TL;DR: This paper proposes fast super-resolution methods while making no compromise on quality, and supports the use of sparse learned dictionaries in combination with neighbor embedding methods, and proposes the anchored neighborhood regression.
Abstract: Recently there have been significant advances in image up scaling or image super-resolution based on a dictionary of low and high resolution exemplars. The running time of the methods is often ignored despite the fact that it is a critical factor for real applications. This paper proposes fast super-resolution methods while making no compromise on quality. First, we support the use of sparse learned dictionaries in combination with neighbor embedding methods. In this case, the nearest neighbors are computed using the correlation with the dictionary atoms rather than the Euclidean distance. Moreover, we show that most of the current approaches reach top performance for the right parameters. Second, we show that using global collaborative coding has considerable speed advantages, reducing the super-resolution mapping to a precomputed projective matrix. Third, we propose the anchored neighborhood regression. That is to anchor the neighborhood embedding of a low resolution patch to the nearest atom in the dictionary and to precompute the corresponding embedding matrix. These proposals are contrasted with current state-of-the-art methods on standard images. We obtain similar or improved quality and one or two orders of magnitude speed improvements.
1,276 citations
TL;DR: An approach to data acquisition, post-processing and visualization that permits the simultaneous non-invasive quantification of multiple important properties of a material or tissue is introduced—which is termed ‘magnetic resonance fingerprinting’ (MRF).
Abstract: Magnetic resonance is an exceptionally powerful and versatile measurement technique. The basic structure of a magnetic resonance experiment has remained largely unchanged for almost 50 years, being mainly restricted to the qualitative probing of only a limited set of the properties that can in principle be accessed by this technique. Here we introduce an approach to data acquisition, post-processing and visualization—which we term ‘magnetic resonance fingerprinting’ (MRF)—that permits the simultaneous non-invasive quantification of multiple important properties of a material or tissue. MRF thus provides an alternative way to quantitatively detect and analyse complex changes that can represent physical alterations of a substance or early indicators of disease. MRF can also be used to identify the presence of a specific target material or tissue, which will increase the sensitivity, specificity and speed of a magnetic resonance study, and potentially lead to new diagnostic testing methodologies. When paired with an appropriate pattern-recognition algorithm, MRF inherently suppresses measurement errors and can thus improve measurement accuracy. A new approach to magnetic resonance, ‘magnetic resonance fingerprinting', is reported, which combines a data acquisition scheme with a pattern-recognition algorithm that looks for the ‘fingerprints’ of interest within the data. Although nuclear magnetic resonance is a powerful analytical tool for many scientific and medical disciplines, usually only a fraction of its potential power is harnessed. Most implementations are qualitative, and restricted in the range of properties that are probed. Dan Ma and colleagues introduce a new approach — termed magnetic resonance fingerprinting — aimed at greatly enhancing the amount of quantitative information that can be obtained in one measurement. Their approach combines a data-acquisition scheme that is indiscriminate in the material properties that it probes with pattern-recognition algorithms that look for the 'fingerprints' of interest within the data. Magnetic resonance fingerprinting has the potential to detect and analyse early indicators of disease or complex changes in materials, as well as increasing the sensitivity, specificity and speed of magnetic resonance studies.
1,253 citations
Proceedings Article•
16 Jun 2013TL;DR: A new general framework for convex optimization over matrix factorizations, where every Frank-Wolfe iteration will consist of a low-rank update, is presented, and the broad application areas of this approach are discussed.
Abstract: We provide stronger and more general primal-dual convergence results for Frank-Wolfe-type algorithms (a.k.a. conditional gradient) for constrained convex optimization, enabled by a simple framework of duality gap certificates. Our analysis also holds if the linear subproblems are only solved approximately (as well as if the gradients are inexact), and is proven to be worst-case optimal in the sparsity of the obtained solutions.
On the application side, this allows us to unify a large variety of existing sparse greedy methods, in particular for optimization over convex hulls of an atomic set, even if those sets can only be approximated, including sparse (or structured sparse) vectors or matrices, low-rank matrices, permutation matrices, or max-norm bounded matrices.
We present a new general framework for convex optimization over matrix factorizations, where every Frank-Wolfe iteration will consist of a low-rank update, and discuss the broad application areas of this approach.
1,246 citations
TL;DR: A very efficient, highly accurate, “Explicit Shape Regression” approach for face alignment that significantly outperforms the state-of-the-art in terms of both accuracy and efficiency.
Abstract: We present a very efficient, highly accurate, "Explicit Shape Regression" approach for face alignment. Unlike previous regression-based approaches, we directly learn a vectorial regression function to infer the whole facial shape (a set of facial landmarks) from the image and explicitly minimize the alignment errors over the training data. The inherent shape constraint is naturally encoded into the regressor in a cascaded learning framework and applied from coarse to fine during the test, without using a fixed parametric shape model as in most previous methods. To make the regression more effective and efficient, we design a two-level boosted regression, shape indexed features and a correlation-based feature selection method. This combination enables us to learn accurate models from large training data in a short time (20 min for 2,000 training images), and run regression extremely fast in test (15 ms for a 87 landmarks shape). Experiments on challenging data show that our approach significantly outperforms the state-of-the-art in terms of both accuracy and efficiency.
1,239 citations
References
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TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Abstract: Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0
18,609 citations
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
Abstract: The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB).
Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising.
BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
9,950 citations
TL;DR: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions, chosen in order to best match the signal structures.
Abstract: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >
9,380 citations
Stanford University1, Cleveland Clinic2, University of Toronto3, Centre national de la recherche scientifique4, Université Paris-Saclay5, University of Paris-Sud6, Avaya7, Rutgers University8, RAND Corporation9, IBM10, University of Pennsylvania11, University of Western Australia12, University of Minnesota13
TL;DR: A publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates is described.
Abstract: The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise, and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate is available, from which we derive a Cp estimate of prediction error; this allows a principled choice among the range of possible LARS estimates. LARS and its variants are computationally efficient: the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.
7,828 citations